\input calc
\def\problemspace{0.4in}

%% TITLE
\noindent
{\bf Math 226: Calculus IV}\hfill
Name:\hbox to 2.5truein{\hrulefill}\medskip\noindent
{\bf Exam III}\quad{\sl April 20, 1993}\hfill
Section:\hbox to 2.5truein{\hrulefill}

\bigskip\noindent
Record your answers to the multiple choice problems by placing an
$\times$ through one letter for each problem on this answer sheet.
There are 15 multiple choice questions worth 6 points each. You
start with 10 points.

\def\answerspace{0.25in}


%% PROBLEM
Compute the Wronskian of the functions $y_1 = e^{-x}\cos(2x)$ and
$ y_2 = e^{-x}\sin(2x)$.

%% CORRECT
$2e^{-2x}$
%% WRONG
$e^{-x}$
%% WRONG
$e^{-2x}(1 + 2\sin(2x)\cos(2x))$
%% WRONG
$e^{2x}(1 - 2\sin(2x)\cos(2x))$
%% WRONG
$-e^{-x}$

%% PROBLEM
Given that $y_1 = e^x$ is a solution of $$x y'' - y' - (x-1)y = 0,
\quad x > 0$$ a second linearly independent solution has the form
$y_2 = v y_1$. Determine which of the following expression gives $v$.

%% CORRECT
$v = \int x e^{-2x}\,dx$
%% WRONG
$v = \int {e^{2x}\over x}\,dx$
%% WRONG
$v = \int {1\over 1+x^2} + 2x\,dx$
%% WRONG
$v = \int \ln(x) - 2x\,dx$
%% WRONG
$v = \int \ln(x-1) - x\,dx$

%% PROBLEM
Let $y(x)$ be the solution to the initial value problem $$y'' - 5
y' - 6y = 0,\quad y(0) = 0,\quad y'(0) = 1$$ Find $y(1)$.

%% CORRECT
${1\over7}(e^6 - e^{-1})$
%% WRONG
$(e^{-2} - e^{-3})$
%% WRONG
$-{1\over5}(e^{-2} + e^{-3})$
%% WRONG
${1\over7}(e - e^{-6})$
%% WRONG
${1\over5}(e^3 + e^2)$

%% PROBLEM
One solution to the differential equation $y'' - 2\alpha y' +
\alpha^2 y = 0$ is $y_1 = e^{\alpha x}$. Find a second linearly
independent solution.

%% CORRECT
$x e^{\alpha x}$
%% WRONG
$e^{\alpha x +1}$
%% WRONG
$e^{-\alpha x}$
%% WRONG
$x^2e^{\alpha x}$
%% WRONG
$e^{\alpha x}\cos(x)$

%% PROBLEM
Find the general solution to the differential equation $y'' + 6y'
+ 10y = 0$.

%% CORRECT
$y = e^{-3x}(c_1\cos(x) + c_2\sin(x))$
%% WRONG
$y = e^{-3x}c_1\cos(2x) + e^{3x}c_2\sin(2x))$
%% WRONG
$y = c_1e^{-4x} + c_2 e^{-2x}$
%% WRONG
$y = c_1e^{-4x} + c_2 e^{2x}$
%% WRONG
$y = c_1e^{-3x} + c_2 xe^{-3x}$

%% PROBLEM
Suppose $y_1 = x$ and $y_2 = x^2$ are solutions to the differential
equation $$y'' + p(x) y' + q(x) y = 0$$ where $p(x)$ and $q(x)$ are
continuous functions for $x > 0$. Find a particular solution to
the non-homogeneous equation $$y'' + p(x)y' + q(x) y = x^3$$

%% CORRECT
$y_p = {1\over12}x^5$
%% WRONG
$y_p = -{1\over4}x^4$
%% WRONG
$y_p = {1\over10}x^5$
%% WRONG
$y_p = {1\over15}x^6$
%% WRONG
$y_p = -{1\over20}x^6$

%% PROBLEM
Using the method of undetermined coefficients, find the correct
form for a particular solution $y_p$ to the differential equation
$$y'' + 12y' + 36 y = x + x^2e^{3x}$$

%% CORRECT
$y_p = (A_0 x + A_1) + (B_0x^2 + B_1 x + B_2)e^{3x}$
%% WRONG
$y_p = A_0 x + x(B_0x^2 + B_1 x + B_2)e^{3x}$
%% WRONG
$y_p = x(A_0 x + A_1) + x^2(B_0x^2 + B_1 x + B_2)e^{3x}$
%% WRONG
$y_p = A x + Bx^2e^{3x}$
%% WRONG
$y_p = (A_0 x + A_1)e^{3x} + (B_0 x + B_1)e^{-6x}$

%% PROBLEM
A mass weighing 1~lb stretches a spring 2~ft.  The mass
is pushed downward, extending the spring an additional 3~ft, and
then released.  Ignore frictional forces and let the downward
direction be positive.  Determine the subsequent displacement $u(t)$
of the mass. ($g = 32 {\rm\  ft}/{\rm sec}^2$).

%% CORRECT
$u(t) = 3 \cos(4t)$
%% WRONG
$u(t) = 3\cos(2t)$
%% WRONG
$u(t) = 3\cos(2t) - {3\over2}\sin(2t)$
%% WRONG
$u(t) = 3\cos(4t) - {1\over4}\sin(4t)$
%% WRONG
$u(t) = 3\cos(t)$

%% PROBLEM
Each of the following is a set of solutions to $y'' - 4y = 0$. Find
the one that is {\bf not} a fundamental set of solutions.

%% CORRECT
$\{e^{2x-1}, e^{2x+1}\}$
%% WRONG
$\{e^{2x} + e^{-2x}, e^{-2x} - e^{-2x}\}$
%% WRONG
$\{e^{1-2x}, e^{1+2x}\}$
%% WRONG
$\{e^{2x}, e^{-2x}\}$
%% WRONG
$\{e^{2x}, -2e^{-2x}\}$

%% PROBLEM
Determine the values of $x$ for which the series $\sum_{n=0}^\infty
{\sqrt{n}\over3^{n+1}}(x-1)^n$ converges.

%% CORRECT
$-2 < x < 4$
%% WRONG
$1-\sqrt{3} < x < 1 + \sqrt{3}$
%% WRONG
$|x| < 3$
%% WRONG
$|x| < \sqrt{3}$
%% WRONG
$|x-1|<1/3$

%% PROBLEM
Compute the Taylor series for $\ln(1 - x^2)$ centered at $x_0 = 0$.

%% CORRECT
$-x^2 - {1\over2}x^4 - {1\over3}x^6 - {1\over4}x^8 - \cdots$
%% WRONG
$x^2 + x^4 + x^6 + x^8 + \cdots$
%% WRONG
$x - x^3 + x^5 - x^7 + \cdots$
%% WRONG
${1\over2}x^2 - {1\over4}x^4 + {1\over6}x^6 - {1\over8}x^8 +\cdots$
%% WRONG
$-x - {1\over3}x^3 - {1\over5}x^5 - {1\over7}x^7 - \cdots$

%% PROBLEM
Find the recurrence relation for the coefficients of a power series
solution centered at $x_0 = 0$ to the differential equation $y''
- xy' - y = 0$.

%% CORRECT
$a_{n+2} = {1\over n+2}a_n$
%% WRONG
$a_{n+2} = {1\over (n+2)(n+1)}a_n$
%% WRONG
$a_{n+2} = -{1\over n+1}a_n$
%% WRONG
$a_{n+2} = {1\over n}a_n$
%% WRONG
$a_{n+2} = -{1\over 2}a_n$

%% PROBLEM
Determine which of the following numbers gives the best lower estimate
for the radius of convergence of a power series solution centered at $x=1$
to the differential equation $(x^2 + 1)y'' - xy' + y = 0$.

%% CORRECT
$\sqrt{2}$
%% WRONG
$1$
%% WRONG
$2$
%% WRONG
$\sqrt{3}$
%% WRONG
$\infty$


%% PROBLEM
Suppose the coefficients of a power series solution, $y(x) =
\sum_{n=0}^\infty a_n x^n$, to a linear differential equation satisfy
the recurrence relation $a_{n+2} = -{1\over n+1} a_n$ for $n \ge 0$.
Find the solution that fulfills the initial conditions $y(0) = 0$,
$y'(0) = 1$.

%% CORRECT
$y = x - {1\over 2}x^3 + {1\over4\cdot 2}x^5 - {1\over6\cdot4\cdot2}x^7 + \cdots$
%% WRONG
$y = x + {1\over3}x^2 + {1\over5}x^4 + {1\over7}x^6 + \cdots$
%% WRONG
$y = x -  {1\over 2}x^3 - {1\over 4}x^5 - {1\over6}x^7 + \cdots$
%% WRONG
$y = x - {1\over2}x^2 + {1\over5\cdot3}x^4 - {1\over7\cdot5\cdot3}x^6 + \cdots$
%% WRONG
$y = x - {1\over2}x^2 + {1\over4\cdot2}x^4 - {1\over6\cdot4}x^6 + \cdots$

%% PROBLEM
Let $p(x)$, $q(x)$, and $g(x)$ be continuous functions of~$x$ and
let $L[y] = y'' + p(x)y' + q(x) y$. Determine which of the following
statements is TRUE.

%% CORRECT
If $y_1$ and $y_2$ are linearly independent solutions to $L[y] =
0$ then any solution to this differential equation can be written
as $c_1y_1 + c_2y_2$ for some choice of constants $c_1$ and $c_2$.
%% WRONG
If $y_1$ and $y_2$ are linearly independent solutions to $L[y] =
g(x)$ then any solution to this differential equation can be written
as $c_1y_1 + c_2y_2$ for some choice of constants $c_1$ and $c_2$.
%% WRONG
$y_1$ and $y_2$ form a fundamental set of solutions to $L[y] = 0$ if and
only if $L[y_1] = 0$ and $L[y_2] = 0$.
%% WRONG
$y_1$ and $y_2$ are linearly independent functions of $x$ if and
only if $W(y_1,y_2) = 0 $.
%% WRONG
If $y_1$ and $y_2$ are solutions to $L[y] = g(x)$ then $c_1y_1 + c_2y_2$ is
also a solution for any choice of constants $c_1$ and $c_2$.

%% END
\end
